Discussiones Mathematicae Graph Theory 30(4) (2010) 555-562
doi: 10.7151/dmgt.1513

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RAINBOW NUMBERS FOR SMALL STARS WITH ONE EDGE ADDED

Izolda Gorgol  and  Ewa Łazuka

Department of Applied Mathematics
Lublin University of Technology
Nadbystrzycka 38D, 20-618 Lublin, Poland
e-mail: i.gorgol@pollub.pl
e-mail: e.lazuka@pollub.pl

Abstract

A subgraph of an edge-colored graph is rainbow if all of its edges have different colors. For a graph H and a positive integer n, the anti-Ramsey number f(n,H) is the maximum number of colors in an edge-coloring of Kn with no rainbow copy of H. The rainbow number rb(n,H) is the minimum number of colors such that any edge-coloring of Kn with rb(n,H) number of colors contains a rainbow copy of H. Certainly rb(n,H) = f(n,H)+1. Anti-Ramsey numbers were introduced by Erdös et al. [5] and studied in numerous papers.

We show that rb(n,K1,4+e) = n+2 in all nontrivial cases.

Keywords: rainbow number, anti-Ramsey number.

2010 Mathematics Subject Classification: 05C55, 05C35.

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Received 29 December 2008
Revised 30 October 2009
Accepted 30 October 2009